Contents

Model selection

from IPython import get_ipython
if get_ipython():
    get_ipython().run_line_magic("load_ext", "autoreload")
    get_ipython().run_line_magic("autoreload", "2")

import numpy as np
import pandas as pd
import torch

import xarray as xr

import matplotlib.pyplot as plt
import seaborn as sns

import collections

import latenta as la
la.logger.setLevel("INFO")

Generative model

n_cells = 1000
cells = la.Dim([str(i) for i in range(n_cells)], "cell")

x1 = la.Fixed(pd.Series(np.random.uniform(0, 1, cells.size), index = cells.index), label = "x1", symbol = "x1")
x1.distribution = la.distributions.Uniform()
x2 = la.Fixed(pd.Series(np.random.uniform(0, 1, cells.size), index = cells.index), label = "x2", symbol = "x2")
x2.distribution = la.distributions.Uniform()

gene_infos = {}
gene_outputs = {}

n_genes = 20
genes = la.Dim(pd.Series([f"constant {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":"constant"} for gene_id in genes.index})

def random_a(n_genes):
    return np.random.choice([-1, 1], n_genes) * np.random.uniform(1., 2., n_genes)
def random_x(n_genes):
    return np.random.uniform(0.15, 0.85, n_genes)

n_genes = 20
gene_type = "linear(x1)"
genes = la.Dim(pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

a = la.Fixed(pd.Series(random_a(n_genes), index = genes.index), label = "a")

gene_outputs[gene_type] = la.links.scalar.Linear(x1, a)

n_genes = 20
gene_type = "switch(x1)"
genes = la.Dim(pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

a = la.Fixed(pd.Series(random_a(n_genes), index = genes.index), label = "a")
switch = la.Fixed(pd.Series(random_x(n_genes), index = genes.index))

gene_outputs[gene_type] = la.links.scalar.Switch(x1, a = a, switch = switch)

n_genes = 20
gene_type = "linear(x2)"
genes = la.Dim(pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

a = la.Fixed(pd.Series(random_a(n_genes), index = genes.index), label = "a")

gene_outputs[gene_type] = la.links.scalar.Linear(x2, a)

n_genes = 20
gene_type = "spline(x1)"
genes = la.Dim(pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

knot = la.Fixed(pd.Series(np.linspace(0., 1., 10), index = pd.Series(range(10), name = "knot")))
a = la.Fixed(pd.DataFrame(np.random.rand(n_genes, knot[0].size) * 2 - 1, columns = pd.Series(range(10), name = "knot"), index = genes.index))

gene_outputs[gene_type] = la.links.scalar.Spline(x1, a = a, knot = knot, smoothness = la.Fixed(10.))

n_genes = 20
gene_type = "linear(x1) + linear(x2)"
genes = la.Dim(pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

a1 = la.Fixed(pd.Series(random_a(n_genes), index = genes.index), label = "a")
a2 = la.Fixed(pd.Series(random_a(n_genes), index = genes.index), label = "a")
gene_outputs[gene_type] = la.modular.Additive(
    x1_effect = la.links.scalar.Linear(x1, a1),
    x2_effect = la.links.scalar.Linear(x2, a2),
    definition = la.Definition([cells, genes])
)

n_genes = 20
gene_type = "linear([x1, x2])"
genes = la.Dim(pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

a = la.Fixed(pd.Series(random_a(n_genes), index = genes.index), label = "a")

gene_outputs[gene_type] = la.links.scalars.Linear([x1, x2], a = a)

n_genes = 20
gene_type = "switch([x1, x2])"
genes = la.Dim(pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

a = la.Fixed(pd.Series(random_a(n_genes), index = genes.index), label = "a")
gene_outputs[gene_type] = la.links.scalars.Linear([
    la.links.scalar.Switch(x1, switch = la.Fixed(pd.Series(random_x(n_genes), index = genes.index))), 
    la.links.scalar.Switch(x2, switch = la.Fixed(pd.Series(random_x(n_genes), index = genes.index)))
], a = a)

n_genes = 20
gene_type = "linear([switch(x1), x2])"
genes = la.Dim(pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene"))
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

a = la.Fixed(pd.Series(random_a(n_genes), index = genes.index), label = "a")
gene_outputs[gene_type] = la.links.scalars.Linear([
    la.links.scalar.Switch(x1, a = la.Fixed(1.), switch = la.Fixed(0.5), label = "X_activity"),
    x2
], a = a)

n_genes = 20
gene_type = "complicated([x1, x2])"
genes.index = pd.Series([f"{gene_type} {i}" for i in range(n_genes)], name = "gene")
gene_infos.update({gene_id:{"type":gene_type} for gene_id in genes.index})

n_knots = [10, 10]
knots_value = np.dstack(np.meshgrid(np.linspace(0, 1, n_knots[0]), np.linspace(0, 1, n_knots[1]))).reshape(-1, 2)
knots_dim = la.Dim(pd.Series(range(knots_value.shape[0]), name = "knot"))
knots = [
    la.Fixed(pd.Series(knots_value[:, 0], index = knots_dim.index)),
    la.Fixed(pd.Series(knots_value[:, 1], index = knots_dim.index))
]
a = la.Fixed(pd.DataFrame(np.random.rand(n_genes, knots_dim.size) * 4 - 1, columns = knots_dim.index, index = genes.index))
smoothness = la.Fixed(10.)
gene_outputs[gene_type] = la.links.scalars.Thinplate(
    [x1, x2], knots = knots, a = a, smoothnesses = smoothness, n_knots = n_knots
)

gene_info = pd.DataFrame.from_dict(gene_infos, orient = "index")
gene_info.index.name = "gene"

genes = la.Dim(gene_info.index)
output = la.modular.Additive(0., la.Definition([cells, genes]), label = "output", subsettable = ("gene",))

for component_id, component in gene_outputs.items():
    setattr(output, component_id, component)

scale = la.Fixed(0.5)

dist = la.distributions.Normal(loc = output, scale = scale)

model_gs = la.Model(dist)
model_gs.plot()

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posterior = la.posterior.Posterior(dist)
posterior.sample(1)

observation_value = posterior.samples[dist].sel(sample = 0).to_pandas()
fig, (ax0, ax1) = plt.subplots(1, 2, figsize = (10, 5))
cell_order = model_gs.find_recursive("x1").prior_pd().sort_values().index
sns.heatmap(observation_value.loc[cell_order], ax = ax0)

<AxesSubplot:xlabel='gene', ylabel='cell'>
../../_images/toy_23_1.png 
output.empirical = observation_value

x1_causal = la.posterior.scalar.ScalarVectorCausal(x1, dist, observed = posterior)
x1_causal.sample(5)
x2_causal = la.posterior.scalar.ScalarVectorCausal(x2, dist, observed = posterior)
x2_causal.sample(5)
x1_x2_causal = la.posterior.scalarscalar.ScalarScalarVectorCausal(x1_causal, x2_causal)
x1_x2_causal.sample(5, n_batch = 40)

gene_ids = gene_info.groupby("type").sample(1).index

x1_causal.plot_features(feature_ids = gene_ids);
x2_causal.plot_features(feature_ids = gene_ids);
x1_x2_causal.plot_features_contour(feature_ids = gene_ids);

/home/wsaelens/projects/probabilistic-cell/latenta/src/latenta/posterior/scalar/scalarscalar/causal.py:297: UserWarning: No contour levels were found within the data range.
  contour = ax.contour(data["x"]["mesh"], data["y"]["mesh"], data["z"]["mesh"], cmap=cmap)
../../_images/toy_27_1.png ../../_images/toy_27_2.png ../../_images/toy_27_3.png

Creating the different models

models = {}

Constant

output.reset_recursive()

mu = la.modular.Additive(intercept = la.Parameter(0., la.Definition([genes])), definition = output.value_definition, subsettable = ("gene",))
s = la.Parameter(1., definition = la.Definition([genes]), transforms = la.distributions.Exponential().biject_to())

mu.x1_effect = la.links.scalar.Constant(x1, output = mu.value_definition)
mu.x2_effect = la.links.scalar.Constant(x2, output = mu.value_definition)

dist = la.distributions.Normal(mu, s)
observation = la.Observation(observation_value, dist, label = "observation")
mu.empirical = observation_value

model_constant = la.Model(observation)
models["constant"] = model_constant
model_constant.plot()

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model_constant2 = model_constant.clone()
assert (model_constant.observation.p.loc.x1_effect.x.loader.value is model_constant2.observation.p.loc.x1_effect.x.loader.value)
assert not (model_constant.observation.p.scale is model_constant2.observation.p.scale)

Additive spline model

model = la.Model(model_constant.observation.clone())

mu = model.observation.p.loc
mu.x1_effect = la.links.scalar.Spline(x1, output = mu)
mu.x2_effect = la.links.scalar.Spline(x2, output = mu)

models["spline(x1) + spline(x2)"] = model

model.plot()

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Thin plate

model = la.Model(model_constant.observation.clone())

mu = model.observation.p.loc
del mu.x1_effect
del mu.x2_effect
mu.x12_effect = la.links.scalars.Thinplate({"x1":x1, "x2":x2}, output = mu)

models["thinplate([x1, x2])"] = model
model.plot()

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Additive switch model

model = la.Model(model_constant.observation.clone())
x1 = model.find_recursive("x1")
x2 = model.find_recursive("x2")

mu = model.observation.p.loc
mu.x1_effect = la.links.scalar.Switch(x1, switch = True, a = True, output = mu.value_definition)
mu.x2_effect = la.links.scalar.Switch(x2, switch = True, a = True, output = mu.value_definition)

models["switch(x1) + switch(x2)"] = model
model.plot()

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Multiplicative switch model

model = la.Model(model_constant.observation.clone())
x1 = model.find_recursive("x1")
x2 = model.find_recursive("x2")

mu = model.observation.p.loc
del mu.x1_effect
del mu.x2_effect
mu.x12_effect = la.links.scalars.Linear([
    la.links.scalar.Switch(x1, switch = True, output = mu), 
    la.links.scalar.Switch(x2, switch = True, output = mu)
], a = la.Definition([genes]))

models["switch([x1, x2])"] = model
model.plot()

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Single and multiplicative switch model with shared switch

model = la.Model(model_constant.observation.clone())
x1 = model.find_recursive("x1")
x2 = model.find_recursive("x2")

mu = model.observation.p.loc
mu.x1_effect = la.links.scalar.Switch(x1, switch = True, a = True, output = mu)
mu.x2_effect = la.links.scalar.Switch(x2, switch = True, a = True, output = mu)
mu.x12_effect = la.links.scalars.Linear([
    la.links.scalar.Switch(x1, switch = mu.x1_effect.switch), 
    la.links.scalar.Switch(x2, switch = mu.x2_effect.switch)
], a = True, output = mu)

models["switch(x1, s1) + switch(x2, s2) + switch([x1, x2], [s1, s2])"] = model
model.plot()

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Linear additive

model = la.Model(model_constant.observation.clone())
x1 = model.find_recursive("x1")
x2 = model.find_recursive("x2")

mu = model.observation.p.loc
mu.x1_effect = la.links.scalar.Linear(x1, a = True, output = mu)
mu.x2_effect = la.links.scalar.Linear(x2, a = True, output = mu)

models["linear(x1) + linear(x2)"] = model
model.plot()

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Additive linear and interaction

model = la.Model(model_constant.observation.clone())
x1 = model.find_recursive("x1")
x2 = model.find_recursive("x2")

mu = model.observation.p.loc
mu.x1_effect = la.links.scalar.Linear(x1, a = True, output = mu)
mu.x2_effect = la.links.scalar.Linear(x2, a = True, output = mu)
mu.x12_effect = la.links.scalars.Linear([x1, x2], a = True, output = mu)

models["linear(x1) + linear(x2) + linear([x1, x2])"] = model
model.plot()

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linear(switch(x1), y)

model = la.Model(model_constant.observation.clone())
x1 = model.find_recursive("x1")
x2 = model.find_recursive("x2")

mu = model.observation.p.loc
del mu.x1_effect
del mu.x2_effect
mu.x12_effect = la.links.scalars.Linear([
    la.links.scalar.Switch(x1, switch = True, label = "x1 activity", symbol = "x1_activity", output = mu), 
    x2
], a = la.Definition([genes]), output = mu)

models["linear([switch(x1), x2])"] = model
model.plot()

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embedding

To illustrate why model selection is necessary, we can also create an extremely flexible embedding model that will certainly overfit. Overfitting in this context means that the embedding will contain information coming from technical noise.

Note that, in a typical use case, we would use an amortization function to infer the latent space, but this is not necessary in this case as we’re simply using a linear function on a simple dataset.

model = la.Model(model_constant.observation.clone())

mu = model.observation.p.loc
# del mu.x1_effect
# del mu.x2_effect

components = la.Dim(20, "component")
embedding = la.Latent(la.distributions.Normal(0., 1.), definition = la.Definition([cells, components]), label = "embedding", symbol = "embedding")
a = la.Latent(la.distributions.Normal(), definition = la.Definition([genes, components]))

mu.embedding_effect = la.links.vector.Matmul(embedding, a)

models["embedding"] = model
model.plot()

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for model_ix, (model_id, model) in enumerate(models.items()):
    print(model_id)
#     if model_id != "constant":
#         continue

    x1 = model.observation.find_recursive("x1")
    x2 = model.observation.find_recursive("x2")
    
#     if "trace" in model: del model["trace"]
    if "trace" not in model:
        inference = la.infer.svi.SVI(model, [la.infer.loss.ELBO()], la.infer.optim.Adam(lr = 0.05))
        trainer = la.infer.trainer.Trainer(inference)
        model["trace"] = trainer.train(1000)
        
        inference = la.infer.svi.SVI(model, [la.infer.loss.ELBO()], la.infer.optim.Adam(lr = 0.01))
        trainer = la.infer.trainer.Trainer(inference)
        model["trace"] = trainer.train(1000)
    
#     if "observed" in model: del model["observed"]
    if "observed" not in model:
        model["observed"] = la.posterior.vector.VectorObserved(model.observation)
        model["observed"].sample(5)
    
#     if "x1_causal" in model: del model["x1_causal"]
    if "x1_causal" not in model:
        x1_causal = la.posterior.scalar.ScalarVectorCausal(x1, model.observation, observed = model["observed"])
        x1_causal.sample(20)
        x1_causal.sample_random(5)
        model["x1_causal"] = x1_causal
        
#     if "x2_causal" in model: del model["x2_causal"]
    if "x2_causal" not in model:
        x2_causal = la.posterior.scalar.ScalarVectorCausal(x2, model.observation, observed = model["observed"])
        x2_causal.sample(20)
        x2_causal.sample_random(5)
        model["x2_causal"] = x2_causal
        
#     if "x1_x2_causal" in model: del model["x1_x2_causal"]
    if "x1_x2_causal" not in model:
        x1_x2_causal = la.posterior.scalarscalar.ScalarScalarVectorCausal(model["x1_causal"], model["x2_causal"])
        x1_x2_causal.sample(20, n_batch = 20)
        model["x1_x2_causal"] = x1_x2_causal

constant

spline(x1) + spline(x2)

thinplate([x1, x2])

switch(x1) + switch(x2)

switch([x1, x2])

switch(x1, s1) + switch(x2, s2) + switch([x1, x2], [s1, s2])

linear(x1) + linear(x2)

linear(x1) + linear(x2) + linear([x1, x2])

linear([switch(x1), x2])

embedding

Explore a model

# model = models["linear(x1) + linear(x2) + linear([x1, x2])"]
# model = models["thinplate([x1, x2])"]
# model = models["constant"]
# model = models["switch(x1) + switch(x2)"]
# model = models["switch([x1, x2])"]
# model = models["spline(x1) + spline(x2)"]
model = models["linear([switch(x1), x2])"]
# model = models["embedding"]

model["x1_causal"].plot_features();
../../_images/toy_60_0.png 
model["x2_causal"].plot_features();
../../_images/toy_61_0.png 
model["x1_x2_causal"].plot_features_contour(feature_ids = gene_ids);
../../_images/toy_62_0.png 
model["x1_x2_causal"].plot_features_contour(feature_ids = gene_info.query("type == 'linear([switch(x1), x2])'").index[:5]);
../../_images/toy_63_0.png 
fig = model["x1_x2_causal"].plot_likelihood_ratio();
fig.axes[0].legend(bbox_to_anchor=(0.5, 1.1), ncol = 2, title = "coregulatory")

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../../_images/toy_64_1.png 
ax = sns.scatterplot(x = "lr_x1", y = "lr_x2", data = model["x1_x2_causal"].scores.join(gene_info), hue = "type")
ax.legend(bbox_to_anchor=(1.1, 1.05))
ax.set_xscale("sigmoid")
ax.set_yscale("sigmoid")
../../_images/toy_65_0.png 
model["x1_x2_causal"].plot_features_contour(
#     feature_ids = gene_info.query("type == 'switch(x1)'").index
);
../../_images/toy_66_0.png

Model selection

model = models["switch(x1) + switch(x2)"]
model = models["switch([x1, x2])"]

# model = models["thinplate([x1, x2])"]
# model["observed"].sample(1)

for model_id, model in models.items():
    print(model_id, model["observed"].elbo.mean().item())

constant -199555.25
spline(x1) + spline(x2) -157754.56872558594
thinplate([x1, x2]) -159050.619140625
switch(x1) + switch(x2) -160976.8098449707
switch([x1, x2]) -161192.58428955078
switch(x1, s1) + switch(x2, s2) + switch([x1, x2], [s1, s2]) -160992.48940980434
linear(x1) + linear(x2) -161196.17388916016
linear(x1) + linear(x2) + linear([x1, x2]) -158545.36004638672
linear([switch(x1), x2]) -168946.1880493164
embedding -171600.1328125

likelihoods = xr.concat([model["observed"].likelihood_features for model in models.values()], dim = pd.Series(models.keys(), name = "model")).to_pandas()
model_ids = [model_id for model_id in likelihoods.index if model_id not in ["embedding"]]
likelihoods = likelihoods.loc[model_ids]
evidences = xr.concat([model["observed"].elbo_features for model in models.values()], dim = pd.Series(models.keys(), name = "model")).to_pandas()
evidences = evidences.loc[model_ids]

sns.heatmap((likelihoods == likelihoods.max(0)).T)

<AxesSubplot:xlabel='model', ylabel='gene'>
../../_images/toy_72_1.png 
sns.heatmap((likelihoods - likelihoods.max(0)).T)

<AxesSubplot:xlabel='model', ylabel='gene'>
../../_images/toy_73_1.png 
sns.heatmap((likelihoods.max(0) - likelihoods).T < 5)

<AxesSubplot:xlabel='model', ylabel='gene'>
../../_images/toy_74_1.png 
# evidences = evidences.loc[[i for i in evidences.index if not i.startswith("thinpl")]]

sns.heatmap((evidences).T)

<AxesSubplot:xlabel='model', ylabel='gene'>
../../_images/toy_76_1.png 
sns.heatmap((evidences.max(0) - evidences).T < 5)

<AxesSubplot:xlabel='model', ylabel='gene'>
../../_images/toy_77_1.png 
sns.heatmap((evidences == evidences.max()).T)

<AxesSubplot:xlabel='model', ylabel='gene'>
../../_images/toy_78_1.png 
selected_evidence = (evidences == evidences.max())

likelihood_diff = (likelihoods - (likelihoods * selected_evidence).min())
undermodelled = (likelihood_diff > 1).any()

gene_info["undermodelled"] = undermodelled

gene_info["selected_evidence"] = selected_evidence.idxmax(0)

gene_info.groupby(["type", "undermodelled"]).count()["selected_evidence"].unstack().T.fillna(0.).style.background_gradient(cmap=sns.color_palette("Blues", as_cmap=True))
type complicated([x1, x2]) constant linear([switch(x1), x2]) linear([x1, x2]) linear(x1) linear(x1) + linear(x2) linear(x2) spline(x1) switch([x1, x2]) switch(x1)
undermodelled                    
False 15.000000 2.000000 20.000000 16.000000 13.000000 15.000000 12.000000 12.000000 20.000000 17.000000
True 5.000000 18.000000 0.000000 4.000000 7.000000 5.000000 8.000000 8.000000 0.000000 3.000000 
pd.crosstab(gene_info["selected_evidence"], gene_info["type"]).style.background_gradient(cmap=sns.color_palette("Blues", as_cmap=True))
type complicated([x1, x2]) constant linear([switch(x1), x2]) linear([x1, x2]) linear(x1) linear(x1) + linear(x2) linear(x2) spline(x1) switch([x1, x2]) switch(x1)
selected_evidence                    
constant 0 20 0 0 0 0 0 2 0 0
linear([switch(x1), x2]) 0 0 20 0 0 0 0 1 0 0
linear(x1) + linear(x2) 0 0 0 0 18 20 17 0 0 0
linear(x1) + linear(x2) + linear([x1, x2]) 0 0 0 20 2 0 3 0 0 0
spline(x1) + spline(x2) 4 0 0 0 0 0 0 13 0 0
switch([x1, x2]) 1 0 0 0 0 0 0 0 20 0
switch(x1) + switch(x2) 1 0 0 0 0 0 0 2 0 13
switch(x1, s1) + switch(x2, s2) + switch([x1, x2], [s1, s2]) 0 0 0 0 0 0 0 2 0 7
thinplate([x1, x2]) 14 0 0 0 0 0 0 0 0 0 
sns.stripplot(likelihood_diff.max(), y = gene_info["type"])

/home/wsaelens/projects/probabilistic-cell/.venv/lib/python3.8/site-packages/seaborn/_decorators.py:36: FutureWarning: Pass the following variable as a keyword arg: x. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  warnings.warn(

<AxesSubplot:ylabel='type'>
../../_images/toy_85_2.png
Model selection Reference